Definition:Constructed Semantics/Instance 1

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Definition

Let $\LL_0$ be the language of propositional logic.

The constructed semantics $\mathscr C_1$ for $\LL_0$ is used for the following results:


Structures

Define the structures of $\mathscr C_1$ as mappings $v$ by the Principle of Recursive Definition, as follows.

Let $\PP_0$ be the vocabulary of $\LL_0$.

Let a mapping $v: \PP_0 \to \set {1, 2}$ be given.

Next, regard the following as definitional abbreviations:

\((1)\)   $:$   Conjunction       \(\ds \mathbf A \land \mathbf B \)   \(\ds =_{\text{def} } \)   \(\ds \map \neg {\neg \mathbf A \lor \neg \mathbf B} \)             
\((2)\)   $:$   Conditional       \(\ds \mathbf A \implies \mathbf B \)   \(\ds =_{\text{def} } \)   \(\ds \neg \mathbf A \lor \mathbf B \)             
\((3)\)   $:$   Biconditional       \(\ds \mathbf A \iff \mathbf B \)   \(\ds =_{\text{def} } \)   \(\ds \paren {\mathbf A \implies \mathbf B} \land \paren {\mathbf B \implies \mathbf A} \)             

It only remains to define $\map v {\neg \phi}$ and $\map v {\phi \lor \psi}$ recursively, by:

\(\ds \map v {\neg \phi}\) \(:=\) \(\ds \begin{cases} 1 & : \text{if } \map v \phi = 2 \\ 2 & : \text{if } \map v \phi = 1 \end{cases}\)
\(\ds \map v {\phi \lor \psi}\) \(:=\) \(\ds \begin{cases} 1 & : \text{if } \map v \phi = \map v \psi = 1 \\ 2 & : \text{otherwise} \end{cases}\)


Validity

Define validity in $\mathscr C_1$ by declaring:

$\models_{\mathscr C_1} \phi$ if and only if $\map v \phi = 2$


Examples


Sources